[Geezer]

Oh, I see where you might be going wrong BC.

It doesn't repeat.
It has no repetition.
Since it happens once, and never again, it does not happen more than once.

The farting does not need to be repeated. We're measuring the effect the farts have on the Earth's rotation in terms of angular displacement in time, not the frequency of farts, and the Earth's angular rotation certainly does repeat.

Please consider this:

Prior to IFD (International Fart Day), the Earth was rotating with uniform angular velocity. If we plot the displacement of a point on the surface near the equator against time, the plot should be perfectly sinusoidal (I hope you would agree with that.)

Likewise, on the days after IFD, the plot will be perfectly sinusoidal with the same frequency and period as the days prior to IFD. (No disagreement so far, I hope.)

During IFD the angular velocity was not uniform. It increased a bit, then it slowed back down so that the daily cycle time was reduced and that resulted in a phase shift of the Earth's rotational cycle relative to our atomic clock "day" (pretty hard to argue with that).

As the angular velocity was not uniform during IFD, the plot of the displacement of the point cannot be perfectly sinusoidal during IFD. In other words, it's distorted. (I'm sure you would agree with that too.)

The only tricky bit is understanding what the distortion from the pure sine means.

I'm sure that the time between two repeating events on the non-sine wave will be less than the time between two repeating events on the pure sine wave, so that alone qualifies as change in frequency. If you look at it in FT terms, I'm pretty confident that will also reveal changes in frequencies.

I'd rather not get into a debate around your cannon ball and chain model until we solve the fart question, but you could post it as a new question.

[JP]

The cosine wave and the sine wave are both easy to do a FT analysis on. Each has exactly one frequency component, and it's the same.

BC, you're also wrong about how to do a Fourier transform of this signal.

1) Your signal isn't a sine + a cosine.  It's a sine over part of the domain, a cosine over another part, and a continuous transition over the third.

2) The FT of a sine or a cosine over a part of the domain is not the same as the FT of a sine or cosine over the whole domain.

3) The FT of the transition region gives you a frequency spread.

4) The FT is linear, so FT(sine bit + cosine bit + transition bit) gets you FT(sine bit)+FT(cosine bit)+FT(transition bit)

5) Because the FT of each bit has multiple frequency components, the entire thing does as well.  I believe the FT(sine bit)+FT(cosine bit)'s multiple components actually cancel each other out if the transition is instantaneous.  But if the transition isn't instantaneous, then I'm extremely confident that you get a spread of frequencies.

By the way, the FT of a sine or a cosine has two frequency components, not one.

By the way, I've done a great deal of work in the area of time-frequency analysis, so I can assure you I know precisely what I'm talking about on this one. 

[Geezer]

It's really just another example of FMF (Fart Modulated Frequency).

[Bored chemist]

JP,
I agree that " if the transition isn't instantaneous, then I'm extremely confident that you get a spread of frequencies".
I'm just saying that, compared to the whole of time, the change is instant.
Taking any other time scale would, as I have said earlier, be an arbitrary choice (and give an equally arbitrary outcome).
If it happened twice then you could (just) use the time between those two events but this cow farting was a one off (I think I may have mentioned that- sorry if I didn't make it clear)
It's the same point I made earlier about apodisation; if you fail to divide by the infinity you don't get zero.(mathematicians of a nervous disposition will want to pretend that I talked about things tending to zero as the reciprocal tends to infinity).
Incidentally, My experience with FT isn't in time/ frequency domain analyses, its in the 2D ones used in optics and the 3D ones used in crystalography, though we do use 1D FTs in spectroscopy.

Geezer,
you say "During IFD the angular velocity was not uniform. It increased a bit, then it slowed back down so that the daily cycle time was reduced and that resulted in a phase shift of the Earth's rotational cycle relative to our atomic clock "day" (pretty hard to argue with that)."
Yes, the earth had an off day in terms of timekeeping.
But only one bad day in the whole of forever. On average, it didn't happen.

"I'd rather not get into a debate around your cannon ball and chain model until we solve the fart question, but you could post it as a new question."
It's much the same system, something gets launched and imparts a torque to the earth, it gets stopped and imparts another, opposite torque. In one case it's a small amount of gas brought to a halt by atmospheric friction, in the other it's a bloody great iron  ball brought to a halt by a chain.
The difference is one of magnitude only. I wanted something  big enough to halt the world for 6 hours and I didn't want to strain the cow fart analogy too far.

[Airthumbs (OP)]

Did someone say that the Earth is a closed system?  Does that exclude, Light, gravity, cosmic rays, dark matter, asteroids, meteors, comets and UFOs?  Not to mention interplanetary dust.  In the cosmic context I do not think earth is a closed system, if it was we could not exist as the dinosaurs would still be running about..  []

[Geezer]

Geezer,
you say "During IFD the angular velocity was not uniform. It increased a bit, then it slowed back down so that the daily cycle time was reduced and that resulted in a phase shift of the Earth's rotational cycle relative to our atomic clock "day" (pretty hard to argue with that)."
Yes, the earth had an off day in terms of timekeeping.
But only one bad day in the whole of forever. On average, it didn't happen.

No argument. It depends on your interpretation of "make the Earth rotate faster". The "faster" bit did happen, just not for very long.

My greater objection was to the point that phase and frequency are independent. That goes against forty years of peering at oscilliscopes.

Understood about the ball and chain thing, but I'm not sure it's any more realistic than the cows. I think I'd prefer some some gigantic rockets. It should not be too difficult to determine how much thrust they would have to develop in order to accelerate the Earth's rotation by a measureable amount.

I'm clueless about how to determine how long the deceleration would take. I suspect the function would be exponential, but how would we calculate the atmospheric friction torque, even to a crude approximation? 

[Geezer]

Wait a minute! BC, you said "On average, it didn't happen".

After the short day, the average absolutely will reveal a difference, unless you are going to count cycles that have not happened yet, so you can't say "it didn't happen", particularly when a permanent change in phase marks the time when it really did happen.

[Bored chemist]

Did someone say that the Earth is a closed system?  Does that exclude, Light, gravity, cosmic rays, dark matter, asteroids, meteors, comets and UFOs?  Not to mention interplanetary dust.  In the cosmic context I do not think earth is a closed system, if it was we could not exist as the dinosaurs would still be running about..  []

We are ignoring quite a lot of things to simplify the model.


Geezer, I'm averaging over an infinite past history. (which is one such simplifying assumption).

[Geezer]

Geezer, I'm averaging over an infinite past history. (which is one such simplifying assumption).

Are you really sure you want to do that? If the Earth doubled its angular velocity tomorrow, would you say it wasn't happening because it had never happened before?

[Bored chemist]

If it did it briefly enough I might.

[JP]

If it only happens once, you can chalk it up to observational error and ignore it, right? 

[Geezer]

If it only happens once, you can chalk it up to observational error and ignore it, right? 

That's what I usually do.

"What the bleep was that?"

"Beats me. Anyway, it doesn't fit the model, so it's obviously a fluke. Ignore it."

Off topic, that's a very common occurrence when testing just about anything with a computer in it. The test engineers see some weird behaviour, but because they can't reproduce the problem, the development engineers tell them they either screwed up or were hallucinating, and close out the problem report.

The weird behaviour usually reappears about two hours before the product is supposed to be released to the market.

[JP]

It's like the old joke about a mathematician, physicist and engineer trying to prove that all odd numbers are primes:

The mathematician says "1 is prime, 3 is prime, 5 is prime, so by induction all odds are primes."

The physicist says "1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 isn't prime (but that's experimental error), 11 is prime, so all odds are prime."

The engineer says "1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, so all odds are prime."

[JP]

Ok, I just came up with this argument that seems to prove by contradiction that "fart day" generates extra frequencies (via the Fourier transform).  Let me know what you think.

First, you need to know that the Fourier transform has an inverse.  From a signal over time, you can uniquely get the frequency spectrum of that signal, and from a frequency spectrum you can uniquely recover the signal over time. 

Second, if the earth rotated unimpeded by farting cows, you could model it by a periodic sinusoid, s(t).  Maybe this sinusoid starts and stops and maybe it goes off to infinity.  It doesn't matter.  It generates a frequency spectrum, say S(f).  You can go back and forth from S(f) to s(t) by Fourier transforms and inverse Fourier transforms.  I could just as easily have told you that the frequency spectrum is S(f) and you could have recovered the signal over time, s(t).  There is no loss of information in the Fourier transform.

Let's assume BC is right and that the frequency spectrum with the cows farting is the same as without.  If that's the case, then it's also given by S(f).  By the properties of the Fourier transform, a frequency spectrum S(f) means that the earth's rotation is given by s(t), which we know is true from above.

But this is identical to the signal without the cows farting, and we know the signals cannot be identical (there's a phase shift).  So the frequency spectra cannot be identical.

QED?

[damocles]

Ok, I just came up with this argument that seems to prove by contradiction that "fart day" generates extra frequencies (via the Fourier transform).  Let me know what you think.

First, you need to know that the Fourier transform has an inverse.  From a signal over time, you can uniquely get the frequency spectrum of that signal, and from a frequency spectrum you can uniquely recover the signal over time. 

Second, if the earth rotated unimpeded by farting cows, you could model it by a periodic sinusoid, s(t).  Maybe this sinusoid starts and stops and maybe it goes off to infinity.  It doesn't matter.  It generates a frequency spectrum, say S(f).  You can go back and forth from S(f) to s(t) by Fourier transforms and inverse Fourier transforms.  I could just as easily have told you that the frequency spectrum is S(f) and you could have recovered the signal over time, s(t).  There is no loss of information in the Fourier transform.

Let's assume BC is right and that the frequency spectrum with the cows farting is the same as without.  If that's the case, then it's also given by S(f).  By the properties of the Fourier transform, a frequency spectrum S(f) means that the earth's rotation is given by s(t), which we know is true from above.

But this is identical to the signal without the cows farting, and we know the signals cannot be identical (there's a phase shift).  So the frequency spectra cannot be identical.

QED?

I think that the problem with this analysis has precisely to do with the phase shift. A frequency spectrum S(f) does not uniquely define a signal s(t) unless we have a firm boundary condition, such as s(0) = 0, To take a simple example the signal function (sin qt + sin 2qt) has precisely the same frequency spectrum as (sin qt + cos 2qt) -- equal spikes at f = q and f = 2q -- but they are quite different functions.

[JP]

To take a simple example the signal function (sin qt + sin 2qt) has precisely the same frequency spectrum as (sin qt + cos 2qt) -- equal spikes at f = q and f = 2q -- but they are quite different functions.

The Fourier spectra of these are actually quite different.   For both signals you get four spikes: at +/-q and +/-2q.  In the first case, they're weighted by i/2, -i/2, i/2, -i/2, respectively.  In the second case, they're weighted by i/2, -i/2, 1/2, 1/2, respectively.  You get spikes at the same frequencies, but the weights are different, so it's a change in Fourier spectrum. 

At some point, it comes down to the definition.  I would say the frequency in the above case has changed, since I'm using the Fourier definition of frequency.  The fact that you get (different) complex weights allows you to invert the transform to get the signals back. 

[Bored chemist]

The FT is an integral (of sorts).

If I integrate something then differentiate it again I lose information because I don't know the "constant of integration".

I think that zero frequency information is lost in the same way.

[JP]

Zero frequency information shouldn't be lost via the FT.  It's what we call a "DC term" in frequency analysis.  If I have a signal s(t)+Constant, then the FT of that is equal to the Fourier transform of s(t) plus a zero-frequency component of amplitude Constant. 

If you invert the Fourier transform, you get exactly the original signal back.

This is because the FT is a definite integral (with infinite limits), and the inverse FT is also a definite integral.  There is no derivative being taken.

The FT and it's inverse do run into some issues with some signals.  I know that figuring out the class of functions it fails on is very difficult, but what was taught to me in physics and optics was that for almost every physical signal, the FT works.  (I do think it has problems representing an instantaneous phase change in a sine wave, for example, but this isn't physical.)

[Geezer]

I do think it has problems representing an instantaneous phase change in a sine wave, for example, but this isn't physical.

I would guess that's because there is "no time" involved, in which case it's no longer a continuous function. It's really a different wave, or a wave that is in two places at the same time, so everything goes haywire.

Presumably that does not happen as soon as you introduce any sort of slope into the function.

[JP]

That's not exactly true.  You can Fourier transform back and forth from a step function or a "top hat" without losing information, even though they're discontinuous.  I could have just done the computation wrong for the sin/cos, which is possible, or there might be something special about it. 

By the way, I'm talking about doing the calculation analytically here.  Of course, trying to do a discrete FT of a step is going to have some issues, since you can't sample the step with perfect resolution.